The Great Depression inhabits modern memory like a malign shadow, always at the edge of our anxieties. Could things ever be that bad again? Are they, perhaps? The cataclysmic poverty and unemployment that swept the world after the Wall Street Crash of 1929 is a nightmare from history that warns that modernity does not always mean progress.

“There’s no way like the American way,” proclaims an upbeat poster photographed by Arthur Rothstein in 1937. Behind it stretches out an ashen wasteland strewn with beaten-up industrial sheds, a battered car and a facade far too sad for Edward Hopper to paint. Yet what finally kicks the last lethal shade of irony into Rothstein’s picture is a black hole that floats in the empty pale sky next to the poster. This deathly disc looks hungry, as if about to swallow the poster, the car, the sheds. It might be the Depression itself: a cancer in time, wasting hope away.

Killed Negatives is an inspired display of accidental surrealism that simultaneously analyses how history is made, and transfigures it into myth.

When we picture the Great Depression we see rural America, even though its effects reached to Wigan Pier and Weimar Berlin. That is because a daring initiative of Roosevelt’s New Deal created some of the most haunting photographs of poverty ever taken. Roy E Stryker of the Farm Security Administration commissioned a small army of photojournalists to travel America portraying the poor, their shattered homes, and a bereft American landscape. The unforgettable results include Walker Evans’s famous portrait of Allie Mae Burroughs and Dorothea Lange’s Migrant Mother.

This small but outstanding exhibition reveals the brutal cropping of history behind those iconic images. Only a minority of photographs satisfied the FSA. Stryker did not simply put the rejects in a filing cabinet. He and his staff punched holes in the negatives so they could not be printed without the defacing presence of a big black hole. They literally “killed” the unwanted pictures.

And what pictures. Stryker had very high standards. What on earth was wrong with Ben Shahn’s compassionate 1935 portrait of a man whose face is full of hunger, not just literal but spiritual, as he drowns in despair? The FSA punched a hole in his chest. It looks as if his heart has been irradiated.

It is hard not to see censorship in some of Stryker’s choices. Many images of black Americans were excluded. In a 1938 shot by John Vachon, a street sign saying “Colored” warns that unemployment was not America’s only problem. In another rejected Rothstein picture, an African-American girl has a hole punched right through her. Did the FSA fear that black faces would fail to elicit compassion?

Yet Stryker seems more like a very sharp picture editor than a shifty censor. His extremely frank letters to the photographer Theodor Jung criticise him for taking too many shots of the same thing, failing to use a tripod and possibly not knowing how to focus his Leica. This is all said in very polite but damning language. The enduring power of the photographs Stryker published proves he knew what he was doing - and he never “killed” a single negative by Lange.

The modern prints made from the FSA’s rejected and damaged negatives bring a new and ghostly vision of history to light. These are not fabrications or jokes. They are real records of a wasted decade. In a 1936 picture by Jung of a farmer breaking down as he discusses his accounts with a man from the FSA, a black hole has materialised between the two men. It seems to represent the gulf between them, the loneliness of the farmer’s despair.

In a 1936 picture by Paul Carter the black disc floats near a wooden house that has tumbled sideways into floodwaters. In a photograph of uncertain date that might be the work of Evans, it hovers between a troubled-looking African American couple on a muggy day as if embodying their worries, stresses and uncertainty. It’s a great photograph, whoever took it – and the circular void adds to its power.

If this archive did not exist, conceptual art would have to invent it. As it is, contemporary artists in the exhibition offer their own takes on these images. Their pieces are a redundant, however, compared with the overwhelming strangeness of the originals. In the same years that Max Ernst and Magritte were trying to release what they called “the marvellous”, the Farm Security Administration created a dream archive of sorrow and anguish, a harrowing and beautiful bluegrass song heard on the 20th century’s lost highway.

Killed Negatives: Unseen Images of 1930s America is at the Whitechapel Gallery, London, until 26 August